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Full text of "Mathematical And Physical Papers - Iii"

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and in general the parts of the force F on which depend the alteration in the time of vibration, and the diminution in the arc of oscillation, vary as ck> a?k', respectively. Hence in the case of a cylinder of small radius, such as the wire used to support a sphere in a pendulum experiment, a considerable change in the radius of the cylinder produces a comparatively small change in the part of the alteration in the time and arc of vibration which is due to the resistance experienced by the wire. The simple formulae (115) are accurate enough for the fine wires usually employed in such experiments if the theory itself be applicable; but reasons will presently be given for regarding the application of the theory to such fine wires as extremely questionable.
From m = "3 or *4 to the end of the table, the first differences of each of the functions m2 (k  1) and mV</ remain nearly constant. Hence for a considerable range of values of m, each of the functions may be expressed pretty accurately by A -f j5m. When m is at all large, the first two terms iu the 2nd and 3rd of the formulae (113) will give k and kr with considerable accuracy, because, independently of the decrease of the successive quantities m"1, m""2, UT3..., the coefficients of m"1 and m"2 are considerably larger than those of several of the succeeding powers. If we neglect in these formulas the terms after ?6.2, we get
k = 1 + /2. ui""1,   k1 = V2 - m"1 + I m'2.
It maybe remarked that these approximate expressions, regarded as functions of the radius a, have precisely the same form as the exact expressions obtained for a sphere, the coefficients only being different.